Revision: Complex Numbers 33 Maths HSC Commerce (English Medium) 11th Standard Maharashtra State Board

z = x + iy, x, y∈ R and \[i=\sqrt{-1}\] is called a complex number. 

x ⇒ Real Part Re(z)

iy ⇒ Imaginary Part Im(z)

If Re(z) = x = 0, then the complex number z is purely imaginary.

If Im(z) =y = 0, then complex number z is purely real.

Integral powers of iota (i):

\[\mathrm{i}^2=-1\]

\[\mathrm{i}^3=-\mathrm{i}\]

\[\mathrm{i}^{4}=1\]

In general,

\[1^{4n}=1\], \[\mathrm{i^{4n+1}=i}\], \[\mathrm{i^{4n+2}=-1}\], \[\mathrm{i^{4n+3}=-i}\] ...where n ∈ N

Conjugate of a complex number z = (a + ib) is defined as \[\overline{z}=\mathrm{a-ib}\].

The cube roots of unity are the solutions of the equation
x³ = 1

They are: 1, \[\frac{-1+i\sqrt{3}}{2}\], \[\frac{-1-i\sqrt{3}}{2}\]

They are denoted by 1, ω, ω²

• Double Conjugate
  z̄̄ = z
• Sum with Conjugate
  z + z̄ = 2 Re(z)
• Difference with Conjugate
  z − z̄ = 2i Im(z)
• Purely Real Condition
  z = z̄ ⇔ z is purely real
• Purely Imaginary Condition
  z + z̄ = 0 ⇔ z is purely imaginary
• Addition
  \[\overline{z_{1}+z_{2}}=\overline{z}_{1}+\overline{z}_{2}\]
• Subtraction
  \[\overline{z_1-z_2}=\overline{z}_1-\overline{z}_2\]
• Multiplication
  \[\overline{z_1z_2}=\overline{z}_1\overline{z}_2\]
• Division
  \[\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z}_1}{\overline{z}_2},z_2\neq0\]
• z · z̄ = [Re(z)]² + [Im(z)]²
• \[\overline{z^{n}}=\left(\overline{z}\right)^{n}\]
• z₁z̄₂ + z̄₁z₂ = 2 Re(z₁z̄₂)

Let √(a + ib) = x + iy

1. Square both sides
   (x + iy)² = a + ib
2. Expand
   x² − y² + 2ixy = a + ib
3. Equate real and imaginary parts
   x² − y² = a
   2xy = b
4. Solve these equations to find x and y
5. Then, √(a + ib) = ±(x + iy)

• ω³ = 1
• 1 + ω + ω² = 0
• ω² = 1/ω
• ω̄ = ω² and \[\left(\overline{\omega}\right)^2=\omega\]
• ω³ⁿ = 1
  ω³ⁿ⁺¹ = ω
  ω³ⁿ⁺² = ω²
• ω + ω² = −1
• ωω² = 1
• arg(ω) = \[\frac{2\pi}{3}\]
  arg(ω²) = \[\frac{4\pi}{3}\]